Problem: $82$ people attended a baseball game. Everyone there was a fan of either the home team or the away team. The number of home team fans was $103$ less than $4$ times the number of away team fans. How many home team and away team fans attended the game?
Let $x$ equal the number of home team fans and $y$ equal the number of away team fans. The system of equations is then: ${x+y = 82}$ ${x = 4y-103}$ Solve for $x$ and $y$ using substitution. Since $x$ has already been solved for, substitute ${4y-103}$ for $x$ in the first equation. ${(4y-103)}{+ y = 82}$ Simplify and solve for $y$ $ 4y-103 + y = 82 $ $ 5y-103 = 82 $ $ 5y = 185 $ $ y = \dfrac{185}{5} $ ${y = 37}$ Now that you know ${y = 37}$ , plug it back into ${x = 4y-103}$ to find $x$ ${x = 4}{(37)}{ - 103}$ $x = 148 - 103$ ${x = 45}$ You can also plug ${y = 37}$ into ${x+y = 82}$ and get the same answer for $x$ ${x + }{(37)}{= 82}$ ${x = 45}$ There were $45$ home team fans and $37$ away team fans.